312 XV NORMED ALGEBRAS AND SPECTRAL THEORY For (i), we have only to copy the proof of (8.3.2), replacing j5f(E; F) by A, and H by G. This proof shows also that the mapping x\-^> x~~ 1 of G into G is differ entiable. Again, the proofs of (ii) and (iii) are identical (apart from nota- tion) with those of (11.1.2) and (11.1.3). (15.2.5) (Gelfand-Mazur theorem) Let A. be a Banach algebra. If A. is afield, then A = Ce. If x e A, there exists A e C such that x — le is not invertible, by virtue of (15.2.4(iii)). But since A is a field, this implies that x = Ae. Hence A = Ce. (15.2.6) Let A. be a Banach algebra with unit element e ^ 0, and let x be an invertible element of A. such that \\x\\ = H*"1!! = 1. Then SpA(x) is contained in the unit circle U. If B is the disk |C| ^ 1 in C, it follows from (15.2.4(iii)) that Sp(x) c B and Sp(x~1) c B. Since Sp^"1) = (Sp^c))'1 by (15.2.3.2), the result follows. (15.2.7) (i) Let A. be a normed algebra. For each xeA, the sequence (\\x?\\1/n) is convergent, and its limit to equctt to inf (||jcn||1/n). n (ii) If A. is a Banach algebra with unit element e ^ 0, then the number p(x) = limdljc"!!17") is equal to the radius of the smallest disk with center 0 which contains SpA(X). (i) Put an = H*"!!. Since the result is obvious if x is nilpotent, we may assume that an > 0 for all n. Clearly we have an+p ^ an ap by virtue of (1 5.1 .1 ). Let m be an integer >0. Then every integer n > 0 is uniquely of the form n = p(ri)m + q(ri), where p(ri) and q(n) are integers and 0 ^ q(n) < m, and we have \imp(ri)/n = l/m. Hence from above we see that an1/n ^ a£(w)/n a^" , H-* 00 and since q(n) takes only m distinct values as n -> oo, it follows that lim sup a*7" ^ a^/m. This is true for all integers m, and therefore n-»-oo lim sup art1/w ^ inf aw1/M ^ lim inf a^", n-*-oo w n-*oo which proves (i) (12.7.11). (ii) If |£|>p(x), then for each r satisfying p(x)<r<\ti\ we have IKC"1*)"!! g(r/|C|)n for all sufficiently large n; hence the series with general term (C""1*)71 is convergent, and its sum is (e-C™"1*)"1 (8.3.2.1). Conse- quently C t SpA(*)- Conversely, if 0 < r < p(x\ the function (e - C'1*)""1 coefficients frt such that J) a«|f«| < + oo. Show that A is a subalgebra of the